the fact that ohthur subjective thought and the objective world are subject to the same laws

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use the decimal system, not because of logical deduction or "freewill," but because we have ten fingers. The word "digital" comes fromthe Latin word for fingers. And to this day, a schoolboy will secretlycount his material fingers beneath a material desk, before arriving at

the answer to an abstract mathematical problem. In so doing, thechild is unconsciously retracing the way in which early humans learnedto count.

The material origins of the abstractions of mathematics were nosecret to Aristotle: "The mathematician," he wrote, "investigatesabstractions. He eliminates all sensible qualities like weight, density,temperature, etc., leaving only the quantitative and continuous (inone, two or three dimensions) and its essential attributes." Elsewhere

he says: "Mathematical objects cannot exist apart from sensible(i.e., material) things." And "We have no experience of anythingwhich consists of lines or planes or points, as we should have if thesethings were material substances, lines, etc., may be prior indefinition to body, but they are not on that account prior insubstance." (1)

The development of mathematics is the result of very material humanneeds. Early man at first had only ten number sounds, precisely

because he counted, like a small child, on his fingers. The exceptionwere the Mayas of Central America who had a numerical systembased on twenty instead of ten, probably because they counted theirtoes as well as their fingers. Living in a simple hunter-gatherersociety, without money or private property, our ancestors had noneed of large numbers. To convey a number larger than ten, hemerely combined some of the ten sounds connected with his fingers.Thus, one more than ten is expressed by "one-ten," (undecim, inLatin, or ein-lifon—"one over"—in early Teutonic, which becomes

eleven in modern English). All the other numbers are onlycombinations of the original ten sounds, with the exception of fiveadditions—hundred, thousand, million, billion and trillion.

The real origin of numbers was already understood by the greatEnglish materialist philosopher of the 17th century Thomas Hobbes:"And it seems, there was a time when those names of number werenot in use; and men were fayn to apply their fingers of one or bothhands, to those things they desired to keep account of; and that

thence it proceeded, that now our numerall words are but ten, in anyNation, and in some but five, and then they begin again." (2)



Alfred Hooper explains: "Just because primitive man invented thesame number of number-sounds as he had fingers, our number-scaletoday is a decimal one, that is, a scale based on ten, and consistingof endless repetitions of the first ten basic number-sounds…Had men

been given twelve fingers instead of ten, we should doubtless have aduo-decimal number-scale today, one based on twelve, consisting ofendless repetitions of twelve basic number-sounds." (3) In fact, aduodecimal system has certain advantages in comparison to thedecimal one. Whereas ten can only be exactly divided by two andfive, twelve can be divided exactly by two, three, four and six.

The Roman numerals are pictorial representations of fingers. Probablythe symbol for five represented the gap between thumb and fingers.

The word "calculus" (from which we derive "calculate") means "pebble"in Latin, connected with the method of counting stone beads on anabacus. These, and countless other examples serve to illustrate howmathematics did not arise from the free operation of the humanmind, but is the product of a lengthy process of social evolution, trialand error, observation and experiment, which gradually becomesseparated out as a body of knowledge of an apparently abstractcharacter. Similarly, our present systems of weights and measureshave been derived from material objects. The origin of the English

unit of measurement, the foot, is self-evident, as is the Spanishword for an inch, "pulgada," which means a thumb. The origin of themost basic mathematical symbols + and – has nothing to do withmathematics. They were the signs used in the Middle Ages by themerchants to calculate excess or deficiency of quantities of goods inwarehouses.

The need to build dwellings to protect themselves from the elementsforced early humans to find the best and most practical way of

cutting wood so that their ends fitted closely together. This meantthe discovery of the right angle and the carpenters’ square. The need

to build a house on level ground led to the invention of the kind oflevelling instrument depicted in Egyptian and Roman tombs, consistingof three pieces of wood joined together in an isosceles triangle, witha cord fastened at the apex. Such simple practical tools were used inthe construction of the pyramids. The Egyptian priests accumulated ahuge body of mathematical knowledge derived ultimately from suchpractical activity.



The very word "geometry" betrays its practical origins. It meanssimply "earth-measurement." The virtue of the Greeks was to give afinished theoretical expression to these discoveries. However, inpresenting their theorems as the pure product of logical deduction,

they were misleading themselves and future generations. Ultimately,mathematics derives from material reality, and, indeed, could have noapplication if this were not the case. Even the famous theorem ofPythagoras, known to every school pupil, that a square drawn on thelongest side of a right triangle is equal to the sum of the squaresdrawn on the other two sides, had been already worked out inpractice by the Egyptians.

Contradictions in Mathematics

Engels, and before him Hegel, pointed to the numerous contradictionsthat abound in mathematics. This was always the case, despite theclaims of perfection and almost papal infallibility made bymathematicians for their "sublime science." This fashion was startedby the Pythagoreans, with their mystical conception of Number, andthe harmony of the universe. Very soon, however, they found outthat their harmonious and orderly mathematical universe was plaguedwith contradictions, the solution of which drove them to despair. For

example, they found that it was impossible to express the length ofthe diagonal of a square in numbers.

The later Pythagoreans discovered that there were many numbers,like the square root of two, which could not be expressed in numbers.It is an "irrational number." But although the square root of twocannot be expressed as a fraction, it is useful to find the length ofthe side of a triangle. Present-day mathematics contains a veritablemenagerie of such strange animals, still untamed, despite all efforts

to domesticate them, but which, once accepted for what they are,render valuable services. Thus we have irrational numbers, imaginarynumbers, transcendental numbers, transfinite numbers, all displayingstrange and contradictory features, and all indispensable to theworkings of modern science.

The mysterious ¹ (pi) was well known to the ancient Greeks, andgenerations of schoolchildren have learned to identify it as the ratiobetween the circumference and diameter of a circle. Yet, strangely,

its exact value cannot be found. Archimedes calculated itsapproximate value by a method known as "exhaustion." It was



between 3.14085 and 3.14286. But if we try to write down theexact value, we get a strange result: ¹ =3.14159265358979323846264338327950…and so on ad infinitum. Pi

(¹) which is now known as a transcendental number, is absolutely

necessary to find the circumference of a circle, but cannot beexpressed as the solution to an algebraic equation. Then we have thesquare root of minus one, which is not an arithmetical number at all.Mathematicians refer to it as an "imaginary number," since no realnumber, when multiplied by itself, can give the result of minus one,because two minuses give a plus. A most peculiar creature, this—butnot a figment of the imagination, despite its name. In Anti-Dühring,Engels points out that:

"It is a contradiction that a negative magnitude should be the squareof anything, for every negative magnitude multiplied by itself gives apositive square. The square root of minus one is therefore not only acontradiction, but even an absurd contradiction, a real absurdity. Andyet Ã –1 is in many cases a necessary result of correct mathematicaloperations. Furthermore, where would mathematics—lower or higher—be, if it were prohibited from operating with Ã –1?" (4) Engels’ remark

is even more true today. This contradictory combination of plus andminus plays an absolutely crucial role in quantum mechanics, where it

appears in a whole host of equations, which are fundamental tomodern science.

That this mathematics involves startling contradictions is not open todoubt. Here is what Hoffman has to say about it:

"That such a formula should have any connection with that world ofstrict experiment which is the world of physics is in itself difficult tobelieve. That it was to be the deep foundation of the new physics,

and that it should actually probe more profoundly than anythingbefore towards the very core of science and metaphysics is asincredible as must once have seemed the doctrine that the earth isround." (5)

Nowadays, the use of the so-called "imaginary" numbers is taken forgranted. The square root of minus one is used for a whole range ofnecessary operations, such as the construction of electrical circuits.Transfinite numbers, in turn, are needed to understand the nature of

time and space. Modern science, and particularly quantum mechanics,could not manage without the use of mathematical concepts which are



frankly contradictory in character. Paul Dirac, one of the founders ofquantum mechanics, discovered the "Q" numbers, which defy the lawsof ordinary mathematics which state that a multiplied by b is thesame thing as b multiplied by a.

Does the Infinite Exist?

The idea of the infinite seems difficult to grasp, because, at firstsight, it is beyond all human experience. The human mind isaccustomed to dealing with finite things, reflected in finite ideas.Everything has a beginning and an end. This is a familiar thought. Butwhat is familiar is not necessarily true. The history of mathematicalthought has some highly instructive lessons on this score. For a long

time, mathematicians, at least in Europe, sought to banish theconcept of infinity. Their reasons for so doing are obvious enough.Apart from the evident difficulty in conceptualising infinity, in purelymathematical terms it involves a contradiction. Mathematics dealswith definite magnitudes. Infinity by its very nature cannot becounted or measured. This means that there is a real conflictbetween the two. For that reason, the great mathematicians ofancient Greece avoided infinity like the plague. Despite this, from thebeginnings of philosophy, men speculated about infinity. Anaximander

(610-547 B.C.) took it as the basis of his philosophy.

The paradoxes of Zeno (c. 450 B.C.) point to the difficulty inherentin the idea of infinitesimal quantity as a constituent of continuousmagnitudes by attempting to prove that movement is an illusion. Zeno"disproved" motion in different ways. He argued that a body inmotion, before reaching a given point, must first have travelled halfthe distance. But before this, it must have travelled half of thathalf, and so on ad infinitum. Thus, when two bodies are moving in the

same direction, and the one behind at a fixed distance from the onein front is moving faster, we assume that it will overtake the other.Not so, says Zeno. "The slower one can never be overtaken by thequicker." This is the famous paradox of Achilles the Swift. Imagine arace between Achilles and a tortoise. Suppose that Achilles can runten times faster than the tortoise which has 1000 metres start. Bythe time Achilles has covered 1000 metres, the tortoise will be 100metres ahead; when Achilles has covered that 100 metres, thetortoise will be one metre ahead; when he covers that distance, the

tortoise will be one tenth of a metre ahead, and so on to infinity.



arithmetic (infinitely large), and successive subdivision in geometry(infinitely small)—he nevertheless polemicised against geometers whoheld that a line segment is composed of infinitely many fixedinfinitesimals, or indivisibles.

This denial of the infinite constituted a real barrier to thedevelopment of classical Greek mathematics. By contrast, the Indianmathematicians had no such scruples and made great advances, which,via the Arabs, later entered Europe. The attempt to banishcontradiction from thought, in accordance with the rigid schemas offormal logic held back the development of mathematics. But theadventurous spirit of the Renaissance opened men’s minds to newpossibilities which were, in truth, infinite. In his book The New

Science (1638), Galileo pointed out that every integer (whole number)has only one perfect square, and every perfect square is the squareof only one positive integer. Thus, in a sense, there are just as manyperfect squares as there are positive integers. This immediately leadsus into a logical contradiction. It contradicts the axiom that thewhole is greater than any of its parts, inasmuch as not all thepositive integers are perfect squares, and all the perfect squaresform part of all the positive integers.

This is only one of the numerous paradoxes which have plaguedmathematics ever since the Renaissance when men began to subjecttheir thoughts and assumptions to a critical analysis. As a result ofthis, slowly, and in the teeth of stubborn resistance fromconservative minds, one by one the supposedly unassailable axioms and"eternal truths" of mathematics have been overthrown. We arrive atthe point where the entire edifice has been shown to be unsound andin need of a thoroughgoing reconstruction on more solid, yet moreflexible foundations, which are already in the process of being laid,

and which will inevitably have a dialectical character.

The Calculus

Many of the so-called axioms of classical Greek mathematics werealready undermined by the discovery of the differential and integralcalculus, the greatest breakthrough in mathematics since the MiddleAges. It is an axiom of geometry that straight and curved areabsolute opposites, and that the two are incommensurable, that is,

the one cannot be expressed in terms of the other. Yet, in the lastanalysis, straight and curved in the differential calculus are regarded



as equal. As Engels points out, the basis for this was laid a long timebefore it was elaborated by Leibniz and Newton: "The turning-pointin mathematics was Descartes’ variable magnitude. With that came

motion and hence dialectics in mathematics, and at once, too, of

necessity the differential and integral calculus, which moreoverimmediately begins, and which on the whole was completed by Newtonand Leibniz, not discovered by them." (7)

The discovery of the calculus opened up a whole new horizon formathematics and science in general. Once the old taboos andprohibitions were lifted, mathematicians were free to investigateentirely new areas. But they made use of infinitely large and smallnumbers uncritically, without considering their logical and conceptual

implications. The use of infinitely small and great quantities wasregarded as a kind of "useful fiction," which, for some reason whichwas not at all clear, always gave the correct result. In the section onQuantity in the first volume of The Science of Logic, Hegel pointsout that, while the introduction of the mathematical infinite openedup new horizons for mathematics, and led to important results, itremained unexplained, because it clashed with the existing traditionsand methods:

"But in the method of the mathematical infinite mathematics finds aradical contradiction to that very method which is characteristic ofitself, and on which it rests as a science. For the calculation of theinfinite admits of, and demands, modes of procedure whichmathematics, when it operates with finite magnitudes, mustaltogether reject, and at the same time it treats these infinitemagnitudes as finite Quanta, seeking to apply to the former thosesame methods which are valid for the latter." (8)

The result was a long period of controversy concerning the validity ofthe calculus. Berkeley denounced it as in open contradiction to thelaws of logic. Newton, who made use of the new method in hisPrincipia, felt obliged to conceal the fact from the public, for fear ofan adverse reaction. In the early 18th century, Bernard Fontenellefinally had the courage to state categorically that inasmuch as thereare infinitely many natural numbers, an infinite number exists as trulyas do finite numbers, and that the reciprocal of infinity is aninfinitesimal. However, he was contradicted by Georges de Buffon,

who rejected the infinity as an illusion. Even the great intellect ofD’Alembert was incapable of accepting this idea. In the article in his



Encyclopaedia on the Differential, he denied the existence of infinity,except in the negative sense of a limit on finite quantities.

The concept of "limit" was in fact introduced in an attempt to get

round the contradiction inherent in infinity. This was especiallypopular in the 19th century, when mathematicians were no longerprepared simply to accept the calculus unthinkingly, as the earliergeneration had been content to do. The differential calculuspostulated the existence of infinitesimally small magnitudes of varyingorders—a first differential, a second differential, and so on toinfinity. By introducing the concept of "limit" they at least createdthe appearance that an actual infinity was not involved. The intentionwas to make the idea of infinity seem subjective, to deny it

objectivity. The variables were said to be potentially infinitely small,in that they become less than any given quantity, as potentiallyinfinite, in that they become larger than any preassigned magnitude.In other words, "as big or small as you like!" This sleight of hand didnot remove the difficulty, but only provided a fig-leaf to cover upthe logical contradictions involved in the calculus.

The great German mathematician Karl Frederick Gauss (1777-1855)was prepared to accept the mathematical infinite, but expressed

horror at the idea of real infinity. However, his contemporaryBernhard Bolanzo, setting out from Galileo’s paradox, began a serious

study of the paradoxes implicit in the idea of a "completed infinite."This work was further developed by Richard Dedekind (1813-1914)who characterised the infinite as something positive, and pointed outthat, in fact, the positive set of numbers can be regarded asnegative (that is, as one that is not infinite). Finally, George Cantor(1845-1918) went far beyond the definition of infinite sets anddeveloped an entirely new arithmetic of "transfinite numbers."

Cantor’s papers, beginning in 1870, are a review of the whole historyof the infinite, beginning with Democritus. Out of this, theredeveloped a whole new branch of mathematics, based on the theoryof sets.

Cantor showed that the points in an area, however large, or in avolume or a continuum of still higher dimension, can be matchedagainst the points on a line or a segment, no matter how small it maybe. Just as there can be no last finite number, so there can be no

last transfinite number. Thus, after Cantor, there can be noargument about the central place of the infinite in mathematics.



Moreover, his work revealed a series of paradoxes which have plaguedmodern mathematics, and have yet to be resolved.

All modern scientific analysis relies on the concept of continuity, that

is to say, that between two points in space, there is an infinitenumber of other points, and also that, between any two points in timethere is an infinite number of other moments. Without making theseassumptions, modern mathematics simply could not function. Yet suchcontradictory concepts would have been indignantly rejected, or atleast regarded with suspicion, by earlier generations. Only thedialectical genius of Hegel (a great mathematician incidentally) wascapable of anticipating all this in his analysis of finite and infinite,space, time and motion.

Yet despite all the evidence, many modern mathematicians persist indenying the objectivity of infinity, while accepting its validity as aphenomenon of "pure" mathematics. Such a division makes no sense atall. For unless mathematics was able to reflect the real, objectiveworld, what use would it be? There is a certain tendency in modernmathematics (and, by extension, incredibly, in theoretical physics) torevert to idealism in its most mystical form, alleging that the validityof an equation is purely a question of its aesthetic value, with no

reference to the material world.

The very fact that mathematical operations can be applied to thereal world and get meaningful results indicates that there is anaffinity between the two. Otherwise, mathematics would have nopractical application, which is clearly not the case. The reason whyinfinity can be used, and must be used, in modern mathematics isbecause it corresponds to the existence of infinity in nature itself,which has imposed itself upon mathematics, like an uninvited guest,

despite all the attempts to bar the door against it.

The reason why it took so long for mathematics to accept infinity wasexplained very well by Engels:

"It is clear that an infinity which has an end but no beginning isneither more or less infinite than one with a beginning but no end.The slightest dialectical insight should have told Herr Dühring thatbeginning and end necessarily belong together, like the North Pole and

the South Pole, and that if the end is left out, the beginning justbecomes the end—the one end which the series has; and vice versa.



The whole deception would be impossible but for the mathematicalusage of working with infinite series. Because in mathematics it isnecessary to start from determinate, finite terms in order to reachthe indeterminate, the infinite, all mathematical series, positive and

negative, must start with 1, or they cannot be used for calculation.But the logical need of the mathematician is far from being acompulsory law for the real world." (9)

Crisis of Mathematics

From our school days we are taught to look upon mathematics, withits self-evident truths "axioms" and its rigorous logical deductions asthe last word in scientific exactitude. In 1900, all this seemed

certain, although in the International Congress of mathematiciansheld that year, David Hilbert set forth a list of the 23 mostsignificant unsolved mathematical problems. From that point thingshave got steadily more complicated, to the point where it is possibleto talk of a real crisis in theoretical mathematics. In his widely-readbook, Mathematics: The Loss of Certainty, published in 1980, MorrisKlein describes the situation thus:

"Creations of the early 19th century, strange geometries and strange

algebras, forced mathematicians, reluctantly and grudgingly, torealise that mathematics proper and the mathematical laws of sciencewere not truths. They found, for example, that several differinggeometries fit spatial experience equally well. All could not be truths.Apparently mathematical design was not inherent in nature, or if itwas, man’s mathematics was not necessarily the account of that

design. The key to reality had been lost. This realisation was thefirst of the calamities to befall mathematics.

"The creation of these new geometries and algebras causedmathematicians to experience a shock of another nature. Theconviction that they were obtaining truths had entranced them somuch that they had rushed impetuously to secure these seemingtruths at the cost of sound reasoning. The realisation thatmathematics was not a body of truths shook their confidence in whatthey had created, and they undertook to reexamine their creations.They were dismayed to find that the logic of mathematics was in sadshape."



At the beginning of the 20th century, they set about trying to solvethe unsolved problems, remove the contradictions, and elaborate anew and foolproof system of mathematics. As Klein explains:

"By 1900 the mathematicians believed they had achieved their goal.Though they had to be content with mathematics as an approximatedescription of nature and many even abandoned the belief in themathematical design of nature, they did gloat over theirreconstruction of the logical structure of mathematics. But beforethey had finished toasting their presumed success, contradictionswere discovered in the reconstructed mathematics. Commonly thesecontradictions were referred to as paradoxes, a euphemism thatavoids facing the fact that contradictions vitiate the logic of

mathematics.

"The resolution of the contradictions was undertaken almostimmediately by the leading mathematicians and philosophers of thetimes. In effect four different approaches to mathematics wereconceived, formulated, and advanced, each of which gathered manyadherents. These foundational schools all attempted not only toresolve the known contradictions but to ensure that no new ones couldever arise, that is, to establish the consistency of mathematics.

Other issues arose in the foundational efforts. The acceptability ofsome axioms and some principles of deductive logic also became bonesof contention on which the several schools took differing positions."

The attempt to eliminate contradictions from mathematics only led tonew and insoluble contradictions. The final blow was struck in 1930,when Kurt Gödel published his famous theorems, which provoked acrisis, even calling into question the fundamental methods of classicalmathematics:

"As late as 1930 a mathematician might perhaps have been contentwith accepting one or another of the several foundations ofmathematics and declared that his mathematical proofs were at leastin accord with the tenets of that school. But disaster struck again inthe form of a famous paper by Kurt Gödel in which he proved, amongother significant and disturbing results, that the logical principlesaccepted by the several schools could not prove the consistency ofmathematics. This, Gödel showed, cannot be done without involving

logical principles so dubious as to question what is accomplished.Gödel’s theorems produced a debacle. Subsequent developments



brought further complications. For example, even the axiomatic-deductive method so highly regarded in the past as the approach toexact knowledge was seen to be flawed. The net effect of thesenewer developments was to add to the variety of possible approaches

to mathematics and to divide mathematicians into an even greaternumber of differing factions." (10)

The impasse of mathematics has produced a number of differentfactions and schools, none of which accept the theories of theothers. There are the Platonists (yes, that’s right), who regard

mathematics as an absolute truth ("God is a mathematician"). Thereare the Conceptualists, whose conception of mathematics is entirelydifferent to that of the Platonists, but it is merely the difference

between objective and subjective idealism. They see mathematics asa series of structures, patterns and symmetries which people haveinvented for their own purposes—in other words, mathematics has noobjective basis, but is purely the product of the human mind! Thistheory is apparently popular in Britain.

Then we have the Formalist school, which was formed at thebeginning of the 20th century, with the specific aim of eliminatingcontradictions from mathematics. David Hilbert, one of the founders

of this school, saw mathematics as nothing more than themanipulation of symbols according to specific rules to produce asystem of tautological statements, which have inner consistency, butotherwise no meaning whatsoever. Here mathematics is reduced to anintellectual game, like chess—again a completely subjective approach.The Intuitionist school is equally determined to separate mathematicsfrom objective reality. A mathematical formula, according to thesepeople, is not supposed to represent anything existing independentlyof the act of computation itself. This has been compared to the

attempt of Bohr to use the discoveries of quantum mechanics tointroduce new views of physical and mathematical quantities asdivorced from objective reality.

All these schools have in common an entirely idealist approach tomathematics. The only difference is that the neo-Platonists areobjective idealists, who think that mathematics originated in the mindof God, and the rest—intuitionists, formalists and conceptualists—believes that mathematics is a subjective creation of the human mind,

devoid of any objective significance. This, then, is the sorry



spectacle presented by the main schools of mathematics in the lastdecade of the 20th century. But it is not the end of the story.

Chaos and Complexity

In recent years, the limitations of mathematical models to expressthe real workings of nature have been the subject of intensediscussion. Differential equations, for example, represent reality as acontinuum, in which changes in time and place occur smoothly anduninterruptedly. There is no room here for sudden breaks andqualitative changes. Yet these actually take place in nature. Thediscovery of the differential and integral calculus in the 18th centuryrepresented a great advance. But even the most advanced

mathematical models are only a rough approximation to reality, validonly within certain limits. The recent debate on chaos and anti-chaoshas centred on those areas involving breaks in continuity, sudden"chaotic" changes which cannot be adequately conveyed by classicalmathematical formulae.

The difference between order and chaos has to do with linear andnon-linear relationships. A linear relationship is one that is easy todescribe mathematically: it can be expressed in one form or another

as a straight line on a graph. The mathematics may be complex, butthe answers can be calculated and can be predicted. A non-linearrelationship, however, is one that cannot easily be resolvedmathematically. There is no straight line graph that will describe it.Non-linear relationships have been historically difficult or impossibleto resolve and they have been often ignored as experimental error.Referring to the famous experiment with the pendulum, James Gleickwrites that the regularity Galileo saw was only an approximation. Thechanging angle of the body’s motion creates a slight non-linearity in

the equations. At low amplitudes, the error is almost non-existent.But it is there. To get his neat results, Galileo also had to disregardnon-linearities that he knew of: friction and air resistance.

Much of classic mechanics is built around linear relationships whichare abstracted from real life as scientific laws. Because the realworld is governed by non-linear relationships, these laws are often nomore than approximations which are constantly refined through thediscovery of "new" laws. These laws are mathematical models,

theoretical constructions whose only justification lies in the insightthey give and their usefulness in controlling natural forces. In the



last twenty years the revolution in computer technology hastransformed the situation by making non-linear mathematicsaccessible. It is for this reason that it has been possible, in anumber of quite separate faculties and research establishments, for

mathematicians and other scientists to be able to do the sums for"chaotic" systems where they could not be done in the past.

James Gleick’s book Chaos, Making a New Science describes how

chaotic systems have been examined by different researchers usingwidely different mathematical models, and yet with all the studiespointing to the same conclusion: that there is "order" in what waspreviously thought of as pure "disorder." The story begins withstudies of weather patterns, in a computer simulation, by an

American meteorologist, Edward Lorenz. Using at first twelve andthen later only three variables in non-linear relationships, Lorenz wasable to produce in his computer a continuous series of conditionsconstantly changing, but literally never repeating the same conditionstwice. Using relatively simple mathematical rules, he had created"chaos."

Beginning with whatever parameters Lorenz chose himself, hiscomputer would mechanically repeat the same calculations over and

over again, yet never get the same result. This "aperiodicity" (i.e.,the absence of regular cycles) is characteristic of all chaoticsystems. At the same time, Lorenz noticed that although his resultswere perpetually different, there was at least the suggestion of"patterns" that frequently cropped up: conditions that approximatedto those previously observed, although they were never exactly thesame. That corresponds, of course, to everyone’s experience of thereal, as opposed to computer-simulated weather: there are"patterns," but no two days or two weeks are ever the same.

Other scientists also discovered "patterns" in apparently chaoticsystems, as widely different as in the study of galactic orbits and inmathematical modelling of electronic oscillators. In these and othercases, Gleick notes, there were "suggestions of structure amidseemingly random behaviour." It became increasingly obvious thatchaotic systems were not necessarily unstable, or could endure for anindefinite period. The well-known "red-spot" visible on the surface ofthe planet Jupiter is an example of a continuously chaotic system

that is stable. Moreover, it has been simulated in computer studiesand in laboratory models. Thus, "a complex system can give rise to



turbulence and cohesion at the same time." Meanwhile, otherscientists used different mathematical models to study apparentlychaotic phenomena in biology. One in particular made a mathematicalstudy of population changes under a variety of conditions. Standard

variables familiar to biologists were used with some of the computedrelationships being, as it would be in nature, non-linear. This non-linearity could correspond, for example, to a unique characteristic ofthe species that might define it as a propensity to propagate, its"survivability."

These results were expressed on a graph plotting the population size,on the vertical axis, against the value of non-linear components, onthe horizontal. It was found that as the non-linearity became more

important—by increasing that particular parameter—so the projectedpopulation went through a number of distinct phases. Below a certaincrucial level, there would be no viable population and, whateverstarting point, extinction would be the result. The line on the graphsimply followed a horizontal path corresponding to zero population.The next phase was a steady state, represented graphically as asingle line in a rising curve. This is equivalent to stable population, ata level that depended on the initial conditions. In the next phasethere were two different but fixed populations, two steady states.

This was shown as a branching on the graph, or a "bifurcation." Itwould be equivalent in real populations to a regular periodic oscillation,in a two year cycle. As the degree of non-linearity increased again,there was a rapid increase in bifurcations, first to a condition whichcorresponded to four steady states (meaning a regular cycle of fouryears), and that very quickly afterwards it was 8, 16, 32, and so on.

Hence, within a short spread of values of the non-linear parameter,a situation had developed which, for all practical purposes, had no

steady state or recognisable periodicity—the population had become"chaotic." It was also found that if the non-linearity was increasedfurther throughout the "chaotic" phase, there would be periods whenapparent steady states returned, based on a cycle of 3 or 7 years,but in each case giving way as non-linearity increased, to furtherbifurcation’s representing 6, 12, and 24 year cycles in the first case,

or 14, 28, and 56 year cycles in the second. Thus, with mathematicalprecision, it was possible to model a change from stability with eithera single steady state or regular, periodic behaviour, to one that was,

for all measurable purposes, random or aperiodic.



This may indicate a possible resolution to debates within the field ofpopulation science between those theorists who believe thatunpredictable population variations are an aberration from a "steadystate norm," and others who believe that steady state is the

aberration from "chaotic norm." These different interpretations mayarise because different researchers have effectively taken a singlevertical "slice" of the rising graph, corresponding to only oneparticular value for non-linearity. Thus, one species could have anorm of a steady or a periodically oscillating population and anothercould exhibit chaotic variability. These developments in biology areanother indication, as Gleick explains, that "chaos is stable; it isstructured." Similar results began to be discovered in a wide varietyof different phenomena. "Deterministic chaos was found in the

records of New York measles epidemics and in 200 years offluctuations of the Canadian lynx population, as recorded by thetrappers of the Hudson’s Bay Company." In all these cases of chaotic

processes, there is exhibited the "period-doubling" that ischaracteristic of this particular mathematical model.

Mandelbrot’s Fractals

Another one of the pioneers of chaos theory, Benoit Mandelbrot, a

mathematician at IBM, used yet another mathematical technique. Inhis capacity as a researcher for IBM, he looked for—and found—"patterns" in a wide variety of natural "random" processes. He found,for instance, that the background "noise" that is always present intelephone transmissions, follows a pattern that is completelyunpredictable, or chaotic, but is nevertheless mathematicallydefinable. Using a computer at IBM, Mandelbrot was able to producechaotic systems graphically, yet only using the simplest mathematicalrules. These pictures, known as "Mandelbrot sets," showed an infinite

complexity, and when a computer drawing was "blown up" to showfiner detail, the vast, seemingly limitless variety continued.

The Mandelbrot sets have been described as possibly the mostcomplex mathematical object or model ever seen. Yet within itsstructure, there were still patterns. By repeatedly "magnifying" thescale and looking at finer and finer detail (something the computercould do indefinitely because the whole structure was based on agiven set of mathematical rules) it could be seen that there were

regular repetitions—similarities—at different scales. "The degree ofirregularity" was the same at different scales. Mandelbrot used the



expression "fractal" to describe the patterns evident within theirregularity. He was able to construct a variety of fractal shapes, byslightly altering the mathematical rules. Thus he was able to producea computer simulation of a coast line which, at any scale (at any

magnification) always exhibited the same degree of "irregularity" or"crinkliness."

Mandelbrot compared his computer-induced systems to examples ofgeometries that were also fractal shapes, repeating the same patternover and over again on different scales. In the so-called MengerSponge, for example, the surface area within it approaches infinity,while the actual volume of the solid approaches zero. Here, it is as ifthe degree of irregularity corresponds to the "efficiency" of the

sponge in taking up space. That may not be as far fetched as it maysound because, as Mandelbrot showed, there are many examples offractal geometry in nature. The branching of the wind-pipe to maketwo bronchiole and their repeated branching right down to the levelof the tiny air passages in the lungs, follows a pattern that can beshown to be fractal. In the same way it can be shown that thebranching of blood vessels is fractal. In other words, there is a"self-similarity," a repeating geometric pattern of branching, atwhatever scale is examined.

The examples of fractal geometry in nature are almost limitless andin his book, The Fractal Geometry of Nature, Mandelbrot sought todemonstrate just that. It has been found that the spectrum of thetiming of a normal heart beat follows fractal laws, perhaps due tothe fractal arrangement of nerve fibres in the heart muscle. Thesame is true of the rapid involuntary eye movements that are afeature of schizophrenia. Thus, fractal mathematics is now routinelyused in a variety of scientific fields, including physiology and

disciplines as widely separated as earthquake studies and metallurgy.

Yet another indications of the deterministic basis of chaos has beenshown in studies of phase transitions and by the use of whatmathematical modellers call "attractors." There are many examples ofphase transitions. It can mean the change from the smooth "laminar"flow of a fluid to turbulent flow, the transition from solid to liquid orliquid to gas, or the change within a system from conductivity to"superconductivity." These phase transitions may have crucial

consequences in technological design and construction. An aircraft,for example would lose lift if the laminar air flow over the wing



became turbulent; likewise, the pressure needed to pump water willdepend on whether or not the flow in the pipe is turbulent.

The use of phase-scale diagrams and attractors represents yet

another mathematical device that has found a wide variety ofapplications in apparently random systems. As in the case of otherchaos studies, there has been the discovery of common patterns, inthis case "strange attractors" in a variety of research programmes,including electric oscillators, fluid dynamics and even in thedistribution of stars in globular clusters. All these variousmathematical devices—period-doubling; fractal geometry; strangeattractors—were developed at different times by differentresearchers to examine chaotic dynamics. But all their results point in

the same direction: that there is an underlying mathematicallawfulness in what was always considered to be random.

A mathematician, Mitchell Feigenbaum, pulling a number of threadstogether, has developed what he has called a "universal theory" ofchaos. As Gleick says "he believed that his theory expressed anatural law about systems at the point of transition between orderand turbulence…his universality was not just qualitative, it was

quantitative…it extended not just to patterns but to precise

numbers."

Marxists would recognise here the similarity with the dialectical lawknown as the law of transformation of quantity to quality. This ideadescribes the transition between one period of more or less gradualdevelopment, when change can be measured or "quantified," and thenext when change has been so "revolutionary," there has been such a"leap," that the entire "quality" of the system has been altered.Gleick’s use of the terms in a similar sense here is yet another

indication of the way modern scientific theory is stumbling towardsmaterialist dialectics.

The central point about the new science is that it deals with theworld as it really is: as a constantly shifting dynamic system.Classical linear mathematics is like formal logic which deals with fixedand unchanging categories. These are good enough as approximations,but do not reflect reality. Dialectics, however, is the logic of change,of processes and as such it represents an advance on formalism. In

the same way, chaos mathematics is a step forward from the rather"unreal" science that ignored uncomfortable irregularities of life.



Quantity and Quality

The idea of the transformation of quantity into quality is implicit inmodern mathematics in the study of continuity and discontinuity. This

was already present in the new branch of geometry, topology,invented in the early years of the 20th century by the great Frenchmathematician, Jules Henri Poincaré (1854-1912). Topology is themathematics of continuity. As Ian Stewart explains it: "Continuity isthe study of smooth, gradual changes, the science of the unbroken.Discontinuities are sudden, dramatic: places where a tiny change incause produces an enormous change in effect." (11)

The standard text-book mathematics gives a wrong impression of how

the world actually is, how nature really works. "The mathematicalintuition so developed," wrote Robert May, "ill equips the student toconfront the bizarre behaviour exhibited by the simplest non-linearsystems." (12) Whereas elementary school geometry teaches us toregard squares, circles, triangles and parallelograms as entirelyseparate things, in topology ("rubber-sheet geometry"), they aretreated as the same. Traditional geometry teaches that the circlecannot be squared, however in topology this is not the case. The rigidlines of demarcation are broken down: a square can be turned

("deformed") into a circle. Despite the spectacular advances of 20thcentury science, it is surprising to note that a large number of whatwould seem to be quite simple phenomena are not properly understoodand cannot be expressed in mathematical terms, for example, theweather, the flow of liquids, turbulence. The shapes of classicalgeometry are inadequate to express the extremely complex andirregular surfaces found in nature, as Gleick points out:

"Topology studies the properties that remain unchanged when shapes

are deformed by twisting or stretching or squeezing. Whether ashape is square or round, large or small, is irrelevant in topology,because stretching can change those properties. Topologists askwhether a shape is connected, whether it has holes, whether it isknotted. They imagine surfaces not just in the one-, two-, andthree-dimensional universes of Euclid, but in spaces of manydimensions, impossible to visualise. Topology is geometry on rubbersheets. It concerns the qualitative rather than the quantitative." (13)

Differential equations deal with the rate of change of position. Thisis more difficult and complex than what may appear at first sight.



Many differential equations cannot be solved at all. These equationsare able to describe motion, but only as a smooth change of position,from one point to another, with no sudden leaps or interruptions.However, in nature, change does not only occur in this way. Periods

of slow, gradual, uninterrupted change are punctuated by sharp turns,breaks in continuity, explosions, catastrophes. This fact can beillustrated by innumerable examples from organic and inorganicnature, the history of society and of human thought. In adifferential equation, time is assumed to be divided into a series ofvery small "time-steps." This gives an approximation of reality, but infact there are no such "steps." As Heraclitus expressed it,"everything flows."

The inability of traditional mathematics to deal with qualitative asopposed to merely quantitative change represents a severe limitation.Within certain limits, it can suffice. But when gradual quantitativechange suddenly breaks down, and becomes "chaotic," to use thecurrent expression, the linear equations of classical mathematics nolonger suffice. This is the starting point for the new non-linearmathematics, pioneered by Benoit Mandelbrot, Edward Lorenz andMitchell Feigenbaum. Without realising it, they were following in thefootsteps of Hegel, whose nodal line of measurement expresses the

very same idea, which is central to dialectics.

The new attitude to mathematics developed as a reaction against thedead end of the existing schools of mathematics. Mandelbrot hadbeen a member of the French school of mathematical Formalismknown as the Bourbaki group, which advocated a purely abstractapproach, proceeding from first principles and deducing everythingfrom them. They were actually proud of the fact that their work hadnothing to do with science or the real world. But the advent of the

computer introduced an entirely new element into the situation. Thisis yet another example of how the development of techniqueconditions that of science. The vast number of computations whichcould be made at the press of a button made it possible to discoverpatterns and lawfulness where previously only random and chaoticphenomena appeared to exist.

Mandelbrot began by investigating unexplained phenomena of thenatural world, like apparently random bursts of interference in radio

transmissions, the flooding of the Nile, and crises of the stockexchange. He realised that the traditional mathematics could not deal



adequately with such phenomena. In investigating infinity in the lastcentury, George Cantor invented the set which is named after him.This involves a line which is divided into an infinite number of points(Cantor "dust") the total length of which is 0. Such a manifest

contradiction disturbed many 19th century mathematicians, yet itserved as the starting point for Mandelbrot’s new theory of fractal

mathematics, which played a key role in chaos theory:

"Discontinuity, bursts of noise, Cantor dusts," Gleick explains, "—phenomena like these had no place in the geometries of the past2,000 years. The shapes of classical geometry are lines and planes,circles and spheres, triangles and cones. They represent a powerfulabstraction of reality, and they inspired a powerful philosophy of

Platonic harmony. Euclid made of them a geometry that lasted twomillennia, the only geometry still that most people ever learn.Aristotle found an ideal beauty in them. But for understandingcomplexity, they turn out to be the wrong kind of abstraction." (14)

All science involves a degree of abstraction from the world of reality.The problem with classical Euclidean measurement, dealing withlength, depth and thickness, is that it failed to capture the essenceof irregular shapes that are found in the real world. The science of

mathematics is the science of magnitude. The abstractions ofEuclidean geometry therefore leave aside all but the quantitative sideof things. Reality is reduced to planes, lines and points. However, theabstractions of mathematics, despite the exaggerated claims madefor them, remain only a rough approximation to the real world, withits irregular shapes and constant and abrupt changes. In the words ofthe Roman poet Horace, "You may drive out nature with a pitch-fork,yet she’ll be constantly running back." James Gleick describes thedifference between classical mathematics and chaos theory in the

following way:

"Clouds are not spheres, Mandelbrot is fond of saying. Mountains arenot cones. Lightning does not travel in a straight line. The newgeometry mirrors a universe that is rough, not rounded, scabrous, notsmooth. It is a geometry of the pitted, pocked, and broken up, thetwisted, tangled, and intertwined. The understanding of nature’s

complexity awaited a suspicion that the complexity was not justrandom, not just accident. It required a faith that the interesting

feature of a lightning bolt’s path, for example, was not its direction,but rather the distribution of zigs and zags. Mandelbrot’s work made



a claim about the world, and the claim was that such odd shapescarry meaning. The pits and tangles are more than blemishesdistorting the classic shapes of Euclidean geometry. They are oftenthe keys to the essence of a thing." (15)

These things were seen as monstrous aberrations by traditionalmathematicians. But to a dialectician, they suggest that the unity offinite and infinite, as in the infinite divisibility of matter, can also beexpressed in mathematical terms. Infinity exists in nature. Theuniverse is infinitely large. Matter can be divided into infinitely smallparticles. Thus, all talk about the "beginning of the universe" and thesearch after the "bricks of matter" and the "ultimate particle" arebased on entirely wrong assumptions. The existence of the

mathematical infinite is merely a reflection of this fact. At the sametime, it is a dialectical contradiction that this infinite universeconsists of finite bodies. Thus, finite and infinite form a dialecticalunity of opposites. The one cannot exist without the other. Thequestion is therefore not whether the universe is finite or infinite. Itis both finite and infinite as Hegel explained long ago.

The advances of modern science have permitted us to penetratedeeper and deeper into the world of matter. At each stage, an

attempt has been made to "call a halt," to erect a barrier, beyondwhich it was allegedly impossible to go. But at each stage, the limitwas overcome, revealing startling new phenomena. Every new andmore powerful particle accelerators have uncovered new and smallerparticles, existing in ever tinier time scales. There is no reason tosuppose that the situation will be any different in relation to thequarks, which at present are being represented as the last of theparticles.

Similarly, the attempt to establish the beginning of the universe and"time" will turn out to be a wild goose chase. There is no limit to thematerial universe, and all efforts to impose one will inevitably fail.The most encouraging thing about the new mathematics of chaostheory is that it represents a rejection of sterile abstractions andivory-tower reductionism, and an attempt to move back towardsnature and the world of everyday experience. And to the degree thatmathematics reflects nature, it must begin to lose its one-sidedcharacter and acquire a whole new dimension which expresses the

dynamic, contradictory, in a word, dialectical character of the realworld.



What is the relation between mathematics and the physical world?For example consider Newton's laws of motion. If we deduce resultsabout mechanics from these laws, are we discovering properties ofthe physical world, or are we simply proving results in an abstract

mathematical system? Does a mathematical model, no matter howgood, only predict behaviour of the physical world or does it give usinsight into the nature of that world? Does the belief that the worldfunctions through simple mathematical relationships tell us somethingabout the world, or does it only tell us something about the wayhumans think. In this article we explore a little of the history of thephilosophy of science in order to look at differing views to the typeof questions that we have just considered.

The most natural starting place historically for examining therelationship between mathematics and the physical world is throughthe views of Pythagoras. The views of Pythagoras are only knownthrough the views of the Pythagorean School for Pythagoras himselfleft no written record of his views. However the views which one hasto assume originated with Pythagoras were extremely influential andstill underlie the today's science. Here we see for the first time thebelief that the physical world may be understood throughmathematics. Music, perhaps strangely, was the motivating factor for

the Pythagoreans realised that musical harmonies were related tosimple ratios. Moreover the same simple ratios hold for vibratingstrings and for vibrating columns of air. The discovery of this generalmathematical principle applying to many apparently differentsituations was seen to be of great significance. Pythagoreans thenlooked for similar mathematical harmonies in the universe in general,in particular the motions of the heavenly bodies. Their belief thatthe Earth is a sphere is almost certainly based on the belief that thesphere was the most perfect solid, so the Earth must be a sphere.

The shadow of the Earth cast on the Moon during an eclipse addedexperimental evidence to the belief.

Plato followed these general principles of Pythagoras and looked foran understanding of the universe based on mathematics. In particularhe identified the five elements, fire, earth, air, water and celestialmatter with the five regular solids, the tetrahedron, cube,octahedron, icosahedron and the dodecahedron. On the one handthere is little merit in Plato's idea: of course Plato's elements are

not the building blocks of matter, and anyway his identification ofthese with the regular solids had little scientific justification. On the

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other hand, at least he was seeking an explanation of the physicalworld using mathematical properties.

Another who followed the general approach of Pythagoras was

Eudoxus. He produced a remarkable model to explain the movementsof the heavenly bodies. It was a remarkable mathematicalachievement based on 27 rotating spheres set up in such a way theretrograde motion of Mars, Jupiter and Saturn were modelled. Herewe have the first major attempt to model the movements of theheavenly bodies, but it is far from clear that Eudoxus thought of hismathematical model as a physical entity. For example he made noattempt to describe the substance of the spheres nor on their modeof interconnection. So it would appear that he thought of his model

as a purely geometric one but the difficulty with the model was thatit did not allow the positions to be predicted with reasonableaccuracy. Despite the fact that the model would not have passed thesimplest of observational tests, Aristotle accepted the crystalspheres of Eudoxus as reality.

Aristotle proposed a scientific method which was highly influential formany centuries. His method, in broad terms, consisted of makingobservations of phenomena, using inductive arguments to deduce

general principles which would explain the observations, then deducingfacts about the phenomena by logical argument from the generalprinciples. He saw this as leading from observations of a fact to anexplanation of that fact. Although Aristotle saw the importance ofnumerical and geometrical relationships in the physical sciences, hemade a very clear distinction between the sciences and puremathematics which he saw as an abstract discipline.

One approach was to set up axioms, that is a list of self-evident

truths, and from these deduce results which were far less obvious.Euclid set up geometry in this way but there were interesting aspectsof this as far as physical science was concerned. On the one handEuclid did not completely achieve his aim, for he did use methods ofproof which went outside his axiom system. In other words he invokedfurther axioms without realising it. More worrying as far as physicalscience was concerned, is the fact that the objects of Euclid'sgeometry can have no physical existence. Points and lines as definedby Euclid could not be physical objects. How can axioms be

considered as self-evident truths when the objects of the axiomshave no physical existence? Archimedes too set up axioms to deduce


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properties of levers. In this he was very successful, for he was ableto create wonderful machines through the understanding that hegained. However, again his axioms refer to objects having propertiesthat no real world object will possess; rods with zero weight, levers

that are perfectly rigid. As a consequence theoretical resultsdeduced from the axioms will never fit experimental evidence exactlybut Archimedes never discussed such points.

Archimedes certainly had developed an excellent mathematical modelbut never discusses its limitations in describing physical situations.Similarly, neither Eudoxus nor Aristotle, despite looking at thephysical reality of the crystal spheres model differently, made clearthe distinction between a mathematical model and reality. The first

to think deeply about this particular problem seems to have beenGeminus. He states clearly that there are two different approachesto modelling the motions of the heavenly bodies, that of the physicistwho looks to explain the motions by the nature of the bodiesthemselves, and the astronomer or mathematician who says that:-

... it is no part of the business of the astronomer to know what is by

nature suited to a position of rest, and what sort of bodies are apt

to move, but he introduces hypotheses under which some bodies

remain fixed, while others move, and then considers to whichhypotheses the phenomena actually observed in the heavens will

correspond.

This approach became known as "saving the appearances", that isputting forward mathematical relationships which correspond toobservation, without making any attempt to suggest a physicalexplanation for the relationships. The most famous of the ancientmodels of the heavenly bodies put forward to "save the appearances"

was that by Ptolemy. His model was the epicycle-deferent modelwhere the motion of the heavenly bodies was circular, but based on anumber of circles whose centres travelled around circles. Ptolemy isquite clear in stating that his model is not intended to representphysical reality, rather it is only a mathematical model that willrepresent what is observed. He also states clearly that othermathematical models are equivalent and will lead to the sameobserved appearance.

The problem of whether a mathematical model represents realitybecame highly significant when Copernicus proposed his Sun centred









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system. The Christian Church had no problems with mathematicalmodels, and were quite happy to allow publication of models to "savethe appearances" based on a Sun centred model. However, this was avery different matter from stating that the Copernican system was

more than a mathematical model, and did indeed represent reality.Clavius, for example, was happy to accept the Copernican model as amathematical model, but he declared that Copernicus had saved theappearances by using axioms which were physically false. Copernicus, however, maintained that his Sun centred system was superior for itprovided an explanation of the retrograde motion of the planets asopposed to Ptolemy's model which was devised to produce theobserved effect.

The strongest supporter of the reality of the Copernican system wasGalileo. He was a great believer in the mathematical approach toscience which originated with the Pythagoreans. For Galileo thesimplicity of the mathematics of the Copernican system over thecomplex mathematics of Ptolemy's system was a strong proof of thereality of the Copernican hypothesis. But it is not only Galileo's beliefin the Copernican system which interests us here, for he made verysignificant advances in understanding the nature of mathematicalmodels. He stressed that an important aspect in understanding

physics is abstraction and idealisation. He could not conductexperiments to test objects falling in a vacuum, nor could he conductexperiments with a pendulum consisting of a point mass supported bya weightless string swinging without air resistance. However, cleverexperiments could lead a scientist towards the idealised situation.Working with the abstract mathematical model of the idealisationwould enable results to be predicted which would be approximatelytrue in reality, and approximate confirmation could be made. Herewas a complete understanding of the relation between the idealised

theory of levers produced by Archimedes so many centuries earlierand real levers. It was a remarkable achievement, but when Galileo was mislead it was often because he had not confirmed an attractivemathematical theory by experiment.

Like Galileo, Kepler believed in the Copernican system. He arguedthat the Sun had a driving force which propelled the planets in theirorbits. This force diminished with distance from the Sun and so theouter planets moved more slowly. Now Kepler could claim that the

Copernican system was real since it provided an explanation for theplanetary motions while that of Ptolemy did not. In Apologia written

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in 1600, but unpublished, Kepler argues that accuracy in "saving thephenomena" cannot distinguish which mathematical theory mightcorrespond to reality. The theory which corresponds to reality willprovide a physical explanation for the appearances. It was a belief

that a simple mathematical relationship must be physically significantwhich led Kepler to discover his third law of planetary motion. Hetried various algebraic formulas to relate the velocity of a planetround the Sun with its distance from the Sun before he stumbled on:

The ratio of the squares of the periods of two planets is directly

proportional to the ratio of the cubes of the radii of their orbits.

The same approach also led him into error. For each of the planets

he calculated1

/√r when r is its distance from the Sun. The numbershe obtained were approximately the densities of materials such asiron, silver and lead. Kepler believed that there must be somephysical significance in this mathematical discovery - of course thereis none.

Another example of a mathematical relation which was thought tohave physical meaning was Bode's law. This took the sequence

4, 4+3, 4+6, 4+12, 4+24, 4+48, 4+96, 4+192, ...divided by 10 to get

0.4, 0.7, 1.0, 1.6, 2.8, 5.2, 10.0, 19.6, ...

Now the distances of the planets Mercury, Venus, Earth, Mars,Jupiter, Saturn from the Sun (taking the distance of the Earth as 1)are

0.39, 0.72, 1.0, 1.52, -, 5.2, 9.5

When Ceres and other asteroids were discovered at distance 2.8 itwas firmly believed that the next planet would be at distance 19.6.When Uranus was discovered at distance 19.2 it was almostconsidered that Bode's law was verified by experiment. However, thenext planets did not fit well at all into the law, though a fewscientists still argue today that Bode's law must be more than amathematical coincidence and result from a physical cause.

[On this theory the outer planets have been disturbed since the













system was created and there is certainly independent evidence thatthis has happened.]

The next axiomatic system we wish to examine is that given by

Newton. He adopted the approach that (see the Principia):-

... particular propositions are inferred from the phenomena, and

afterwards rendered general by induction. Thus it was that the

impenetrability, the mobility, the impulsive force of bodies, and the

laws of motion and of gravitation, were discovered.

He set up an axiom system consisting of hard particles which were atrest or in motion, obeying three simple laws concerning motion and

forces, and a universal law of gravitation. Newton was careful todistinguish between laws which he believed he had verified, andunderlying reasons why the laws existed. For example he believed hehad proved his law of gravitation, but he was clear that he putforward no explanation of why or how two bodies underwent mutualattraction in a vacuum. One could argue that Newton was "saving theappearances" again, putting forward a mathematical model of theworld without any physical explanations. He did, however, make veryclear the relationship between mathematical dynamical results proved

from his axioms and the outcomes of experiments conducted in thereal world.

As we have suggested there were problems with Newton's systemdespite the fact that it appeared to reduce the whole of nature toconsequences of simple mathematical laws. Perhaps most significantwas the fact that his theories required a postulate of absolute spaceand time. He was well aware of this and he put forward his rotatingbucket experiment to try to prove that absolute space did exist. But

there was a weakness here, namely he had introduced a concept ofspace independent of the material of the universe. Is space anindependent concept, or are there simply relations between thematerial objects? Berkeley criticised Newton's absolute space byasking how spatial relationships could be meaningful in a world withoutmatter. If there is only one particle in the universe, said Berkeley, how is it meaningful to say that it is at rest or, for that matter,what could it possibly mean to say that it was accelerating.

Although Newton had made a clear distinction between amathematical theory and a physical reality, Berkeley argued that he












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had fallen into his own trap for he spoke of forces as physicalentities, where Berkeley believed that they were nothing other thanterms in the equations set up by Newton. Indeed Berkeley arguedagainst abstract ideas in general for, in his view, they led to the

mistaken belief in the reality of concepts such as force, absolutespace, absolute time, and absolute motion. No idea, argued Berkeley, can exist unperceived and nothing exists except things which areperceived.

Poincaré put forward important ideas on mathematical models of thereal world. If one set of axioms is preferred over another to model aphysical situation then, Poincaré claimed, this was nothing more thana convention. Conditions such as simplicity, easy of use, and

usefulness in future research, help to determine which will be theconvention, while it is meaningless to ask which is correct. Thequestion of whether physical space is euclidean is not a meaningfulone to ask. The distinction, he argues, between mathematicaltheories and physical situations is that mathematics is a constructionof the human mind, whereas nature is independent of the humanmind. Here lies that problem; fitting a mathematical model to realityis to forcing a construct of the human mind onto nature which isultimately independent of mind.

Article by: J J O'Connor and E F Robertson

Mathematics has been called the language of the universe. Scientistsand engineers often speak of the elegance of mathematics whendescribing physical reality, citing examples such as π, E=mc2, andeven something as simple as using abstract integers to count real-world objects. Yet while these examples demonstrate how useful mathcan be for us, does it mean that the physical world naturally follows

the rules of mathematics as its "mother tongue," and that thismathematics has its own existence that is out there waiting to bediscovered? This point of view on the nature of the relationshipbetween mathematics and the physical world is called Platonism, butnot everyone agrees with it.

Derek Abbott, Professor of Electrical and Electronics Engineering atThe University of Adelaide in Australia, has written a perspectivepiece to be published in the Proceedings of the IEEE in which he

argues that mathematical Platonism is an inaccurate view of reality.Instead, he argues for the opposing viewpoint, the non-Platonist









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notion that mathematics is a product of the human imagination thatwe tailor to describe reality.

This argument is not new. In fact, Abbott estimates (through his own

experiences, in an admittedly non-scientific survey) that while 80% ofmathematicians lean toward a Platonist view, engineers by and largeare non-Platonist. Physicists tend to be "closeted non-Platonists," hesays, meaning they often appear Platonist in public. But when pressedin private, he says he can "often extract a non-Platonist confession."

So if mathematicians, engineers, and physicists can all manage toperform their work despite differences in opinion on this philosophicalsubject, why does the true nature of mathematics in its relation to

the physical world really matter?The reason, Abbott says, is that because when you recognize thatmath is just a mental construct— just an approximation of reality thathas its frailties and limitations and that will break down at some pointbecause perfect mathematical forms do not exist in the physicaluniverse—then you can see how ineffective math is.

And that is Abbott's main point (and most controversial one): thatmathematics is not exceptionally good at describing reality, anddefinitely not the "miracle" that some scientists have marveled at.Einstein, a mathematical non-Platonist, was one scientist whomarveled at the power of mathematics. He asked, "How can it bethat mathematics, being after all a product of human thought whichis independent of experience, is so admirably appropriate to theobjects of reality?"

In 1959, the physicist and mathematician Eugene Wigner describedthis problem as "the unreasonable effectiveness of mathematics." Inresponse, Abbott's paper is called "The Reasonable Ineffectivenessof Mathematics." Both viewpoints are based on the non-Platonist ideathat math is a human invention. But whereas Wigner and Einsteinmight be considered mathematical optimists who noticed all the waysthat mathematics closely describes reality, Abbott pessimisticallypoints out that these mathematical models almost always fall short.

What exactly does "effective mathematics" look like? Abbott explainsthat effective mathematics provides compact, idealizedrepresentations of the inherently noisy physical world.

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"Analytical mathematical expressions are a way making compactdescriptions of our observations," he told Phys.org . "As humans, wesearch for this 'compression' that math gives us because we havelimited brain power. Maths is effective when it delivers simple,

compact expressions that we can apply with regularity to manysituations. It is ineffective when it fails to deliver that elegantcompactness. It is that compactness that makes it useful/practical... if we can get that compression without sacrificing too muchprecision.

"I argue that there are many more cases where math is ineffective(non-compact) than when it is effective (compact). Math only has theillusion of being effective when we focus on the successful examples.

But our successful examples perhaps only apply to a tiny portion of allthe possible questions we could ask about the universe."

Some of the arguments in Abbott's paper are based on the ideas ofthe mathematician Richard W. Hamming, who in 1980 identified fourreasons why mathematics should not be as effective as it seems.Although Hamming resigned himself to the idea that mathematics isunreasonably effective, Abbott shows that Hamming's reasonsactually support non-Platonism given a reduced level of mathematical

effectiveness.

Here are a few of Abbott's reasons for why mathematics isreasonably ineffective, which are largely based on the non-Platonistviewpoint that math is a human invention:

• Mathematics appears to be successful because we cherry-pick theproblems for which we have found a way to apply mathematics. Therehave likely been millions of failed mathematical models, but nobody

pays attention to them. ("A genius," Abbott writes, "is merely onewho has a great idea, but has the common sense to keep quiet abouthis other thousand insane thoughts.")

• Our application of mathematics changes at different scales. For

example, in the 1970s when transistor lengths were on the order ofmicrometers, engineers could describe transistor behavior usingelegant equations. Today's submicrometer transistors involvecomplicated effects that the earlier models neglected, so engineers

have turned to computer simulation software to model smaller

the fact that ohthur subjective thought and the objective world are subject to the same laws

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